S-Duality for D3-Brane in NS-NS and R-R Backgrounds

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1 S-Duality for D3-Brane in NS-NS and R-R Backgrounds Chen-Te Ma Collaborator: Pei-Ming Ho National Taiwan University arxiv: [hep-th] January 17, 2014

2 Reference P. -M. Ho and Y. Matsuo, M5 from M2, JHEP 0806 (2008) 105 [arxiv: [hep-th]]. P. -M. Ho, Y. Imamura, Y. Matsuo and S. Shiba, M5-brane in three-form flux and multiple M2-branes, JHEP 0808 (2008) 014 [arxiv: [hep-th]]. P. -M. Ho and C. -H. Yeh, D-brane in R-R Field Background, JHEP 1103 (2011) 143 [arxiv: [hep-th]]. P. -M. Ho and C. -T. Ma, Effective Action for Dp-Brane in Large RR (p-1)-form Background, JHEP 1305 (2013) 056 [arxiv: [hep-th]]. P. -M. Ho and C. -T. Ma, S-Duality for D3-Brane in NS-NS and R-R Backgrounds, arxiv: [hep-th].

3 From NS-NS D3 to R-R D3 M5 R-R D4 NS-NS D4 D3

4 From NS-NS D3 to R-R D3 D3 Field Redefinition, NS-NS D3 R-R D3

5 Poisson Limit L NS NS 1 4 F αβ F αβ 1 2 F α µf α µ 1 4 F µ νf µ ν. F AB F AB + g{a A, a B }.

6 Poisson Limit L NS NS 1 4 F αβ F αβ 1 2 F α µf α µ 1 4 F µ νf µ ν. F AB F AB + g{a A, a B }. b µ ɛ µ ν a ν. F = F +, } = g{a 1 a 2 H 1 2.

7 Poisson Limit L NS NS 1 4 F αβ F αβ 1 2 F α µf α µ 1 4 F µ νf µ ν. F AB F AB + g{a A, a B }. b µ ɛ µ ν a ν. F = F +, } = g{a 1 a 2 H 1 2. F α µ ɛ αβ F β ν ɛ ν µ = ɛ αβ ( β b µ B β ν V ν µ ).

8 Poisson Limit L NS NS 1 4 F αβ F αβ 1 2 F α µf α µ 1 4 F µ νf µ ν. F AB F AB + g{a A, a B }. b µ ɛ µ ν a ν. F = F +, } = g{a 1 a 2 H 1 2. F α µ ɛ αβ F β ν ɛ ν µ = ɛ αβ ( β b µ B β ν V ν µ ). L NS NS = 1 4 F αβ F αβ F α µf α µ 1 2 H 1 2 H 1 2.

9 Poisson Limit L (1) NS NS 1 2 φ ɛαβ F αβ φ F α µf α µ 1 2 H 1 2 H 1 2.

10 Poisson Limit L (1) NS NS 1 2 φ ɛαβ F αβ φ F α µf α µ 1 2 H 1 2 H 1 2. φ = F 01.

11 Poisson Limit L (1) NS NS 1 2 φ ɛαβ F αβ φ F α µf α µ 1 2 H 1 2 H 1 2. φ = F 01. L (2) NS NS = 1 2 F ɛαβ F αβ F F α µf α µ 1 2 H 1 2 H 1 2. ) µ (F 1 2 F 01 = 0.

12 Poisson Limit F 01F 1 2 = F 01F {a 0, a 1}F 1 2 = ɛ αβ β a µ B α µ + ɛ αβ F 1 2 B α 1 B 2 β + total derivatives.

13 Poisson Limit F 01F 1 2 = F 01F {a 0, a 1}F 1 2 = ɛ αβ β a µ B α µ + ɛ αβ F 1 2 B α 1 B 2 β + total derivatives. ɛ αβ f β µ [ B α µ ɛ µ ν ν a α].

14 Poisson Limit F 01F 1 2 = F 01F {a 0, a 1}F 1 2 = ɛ αβ β a µ B α µ + ɛ αβ F 1 2 B α 1 B 2 β + total derivatives. ɛ αβ f β µ [ B α µ ɛ µ ν ν a α]. If we integrate out a α, we get ɛ µ ν ν f β µ = 0. It implies that locally f β µ = µ a β for some field a β.

15 Poisson Limit F 01F 1 2 = F 01F {a 0, a 1}F 1 2 = ɛ αβ β a µ B α µ + ɛ αβ F 1 2 B α 1 B 2 β + total derivatives. ɛ αβ f β µ [ B µ α ɛ µ ν ν a α]. If we integrate out a α, we get ɛ µ ν ν f β µ = 0. It implies that locally f β µ = µ a β for some field a β. ɛ αβ µ a β B µ α.

16 Poisson Limit We can now easily integrate out B α µ. The result is equivalent to replacing B α µ by the solution of its equation of motion.

17 Poisson Limit We can now easily integrate out B α µ. The result is equivalent to replacing B α µ by the solution of its equation of motion. F 01F g ɛαβ F αβ F α µ = F α µ.

18 Poisson Limit We can now easily integrate out B α µ. The result is equivalent to replacing B α µ by the solution of its equation of motion. F 01F g ɛαβ F αβ F α µ = F α µ. L RR = 1 2 H F α µf α µ 1 4 F µ νf µ ν + 1 2g ɛαβ F αβ.

19 Poisson Limit We can now easily integrate out B α µ. The result is equivalent to replacing B α µ by the solution of its equation of motion. F 01F g ɛαβ F αβ F α µ = F α µ. L RR = 1 2 H F α µf α µ 1 4 F µ νf µ ν + 1 2g ɛαβ F αβ. Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.

20 Coupling Constant L NS NS 1 g G 2 [ 1 4 F αβ F αβ 1 2 F α µf α µ 1 4 F µ νf µ ν], where g G is the gauge coupling.

21 Coupling Constant L NS NS 1 g G 2 [ 1 4 F αβ F αβ 1 2 F α µf α µ 1 4 F µ νf µ ν], where g G is the gauge coupling. b µ g G 2 b µ, B α µ g G 2 B α µ, a µ g G 2 a µ, a α g G 2 a α.

22 Coupling Constant L RR 1 [ gg H F α µf α µ 1 4 F µ νf µ ν + 1 ] 2θ ɛαβ F αβ, but with the gauge coupling g G and noncommutativity parameter θ defined by g G 1 g G, θ g G 2 θ, and with all the coupling constants g replaced by θ.

23 Coupling Constant Notice that the need of two independent parameters (g G, θ ) or (g G, θ) can be seen only if higher order terms are included. At the lowest order, we can scale the gauge fields so that the R-R action has g G = g G and θ = θ.

24 Double Scaling Limit The double scaling limit of NS-NS is: l s ɛ 1/4, g s ɛ 1/2, B µ ν 1, g αβ 1, g µ ν ɛ.

25 Double Scaling Limit The double scaling limit of NS-NS is: l s ɛ 1/4, g s ɛ 1/2, B µ ν 1, g αβ 1, g µ ν ɛ. The double scaling limit of R-R is: l s ɛ 1/2, g s ɛ 1/2, B µ ν 1, g αβ 1, g µ ν ɛ.

26 Double Scaling Limit The double scaling limit of NS-NS is: l s ɛ 1/4, g s ɛ 1/2, B µ ν 1, g αβ 1, g µ ν ɛ. The double scaling limit of R-R is: l s ɛ 1/2, g s ɛ 1/2, B µ ν 1, g αβ 1, g µ ν ɛ. Despite the fact that the string coupling g s is large for the R-R theory when it is small for the NS-NS theory, the decoupling of the D3-brane from the bulk in the Seiberg-Witten limit ensures that its S-dual picture also has an effective world-volume theory decoupled from the bulk.

27 S-Duality to all Orders F AB F AB + [a A, a B ]. [A, B] A B B A. ( gɛ µ ν µ ν A B A exp 2 ) B.

28 S-Duality to all Orders F AB F AB + [a A, a B ]. [A, B] A B B A. ( gɛ µ ν µ ν A B A exp 2 ) B. F 1 2 = H 1 2, d 4 x 1 2 ɛαβ F αβ F 1 2 = d 4 x 1 2g ɛαβ F αβ, ɛ βα ɛ µ ν F β ν = F α µ.

29 S-Duality to all Orders H 1 2 = H [b 1, b 2 ], F 1 2 = F 1 2, ) ɛ αβ F β µ = ( α b µ g{ ρ X µ, ˆB ρ α }, F αβ = F αβ + g[ F α µ ˆB µ β + F β µ ˆB µ α ] + g 2 F µ ν {ˆB µ α, ˆB ν β }.

30 S-Duality to all Orders H 1 2 = H [b 1, b 2 ], F 1 2 = F 1 2, ) ɛ αβ F β µ = ( α b µ g{ ρ X µ, ˆB ρ α }, F αβ = F αβ + g[ F α µ ˆB µ β + F β µ ˆB µ α ] + g 2 F µ ν {ˆB µ α, ˆB ν β }. as (A B) A exp( gɛ µ1 ν1 µ1 ν1 2 ) 1 gɛ µ B 2 ν 2 µ2 ν2 {A, B} (A B) + (B A).

31 Gauge Algebra and Closedness {A, {B, C} } = {{A, B}, C}, although it is not associative without integration.

32 Gauge Algebra and Closedness {A, {B, C} } = {{A, B}, C}, although it is not associative without integration. gɛ µ ν { µ f, ν g} = [f, g].

33 Conclusion and Future We extend S-duality to infinite order. We also find non-commutative R-R D3-brane beyond Poisson. The gauge algebra of this theory is also extended to all orders.

34 Conclusion and Future We extend S-duality to infinite order. We also find non-commutative R-R D3-brane beyond Poisson. The gauge algebra of this theory is also extended to all orders. DBI?

S-Duality for D3-Brane in NS-NS and R-R Backgrounds

S-Duality for D3-Brane in NS-NS and R-R Backgrounds arxiv:1311.3393v3 [hep-th] 15 Jan 2014 Pei-Ming Ho a,b,c,1 and Chen-Te Ma a,2 a Department of Physics and Center for Theoretical Sciences, b Center for